In particular if the equation has two independent variables then it can be transformed in an ordinary differential equation, moreover if an ordinary differential equation is also invariant under an one-parameter group, then its order can be reduced in one, in the case of first order ordinary equations, the symmetry group can be used to find a solution in terms of cuadratures by two very well known methods, canonical coordinates and the Lie integrating factor.

These conditions permit to reduce the nonlinear coupled two-dimensional problem to the two-point boundary value problem for the second-order nonlinear ordinary differential equation, and then, to a nonlinear equation, with respect to sludge concentration.

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